Concise Physical Chemistry

c09 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

132 THE PHASE RULE

The definition ofVm 2 should come as no surprise. It comes from the condition on

perfect differentials. In the case of the volume, we have

dV=

(

∂V

∂p

)

T,ni,nj

dp+

(

∂V

∂T

)

p,ni,nj

dT+

∑

ni

(

∂V

∂ni

)

T,p,nj

dni

This generalization involving the sum

∑

ni

(

∂V

∂ni

)

T,p,nj

dni

gets us away from the restriction that the solution be binary, a condition we imposed

at the beginning for simplicity. Now the concepts developed can be applied to any

number of components. Recalling that volume is a thermodynamic property, we have

V=f(p,T,n 1 ,n 2 ,...,nj)

In view of the first two terms in the sum, the functions are sensitive to variations in

both pressure and temperature hence one or both may be held constant in the phase

diagrams discussed below. The variation of total volume with composition in curves

like Fig. 9.7 gives rise to the termexcess volumeas the volume above the straight

line expected from a simple sum of molar volumes in the pure stateVm◦. The excess

volume can be negative leading to a nonideal curve below the straight line in Fig. 9.7.

The volume in a real binary system corresponds to the sum in which

V=Vm◦ 1 n 1 +Vm◦ 2 n 2

is replaced by

V=V ̄m 1 n 1 +V ̄m 2 n 2

where the molar volumesV ̄m 1 andV ̄m 2 are no longer volumes in the pure phase but

arepartial molar volumes, unique to the ratio ofn 1 andn 2 in the solution. In general,

V ̄mi=

(

∂V

∂ni

)

T,p,nj

Each partial molar volume must be determined experimentally. There are, of course,

simplified equations containing empirical constants that work more or less well for

real (nonideal) solutions just as there are for nonideal gases.

The greatest generalization in this field, however, is to recognize that nothing in

the arguments made is specific to volume alone. The general rule is thatpartial molar